Finite Group Databasebeta

← #13053638

Subgroups — #13053638

Order 1344.

292 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621551C211yesmaximal, characteristic, complemented
2C7×#908C211yesmaximal, characteristic, complemented
3#10620932C211yesmaximal, characteristic, complemented
4#10620866C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C3×#6398C211yesmaximal, characteristic, complemented
7#10620513C211yesmaximal, characteristic, complemented
8#89973—31nomaximal, Hall
9#66728C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66686—22no—
14D21⋊Q8—21no—
15#66673—21no—
16#66672—21no—
17D3×Dic14—21no—
18C3×C7×M16C2211yescharacteristic
19C4×Dic21C2211yescharacteristic
20C7×D24—21no—
21C7×#204—21no—
22C3×#1136C2211yescharacteristic
23(C2×D3)⋊Dic7—22no—
24#66567—21no—
25#66562—21no—
26#6554—31no—
27C7×AGL(1,ℤ/8)—31no—
28#6499—31no—
29#6495—31no—
30#6411—31no—
31#6399—31no—
32#6398D311yescharacteristic, complemented
33#4753—71nomaximal, Hall
34C22×D3×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37C2×Dic21—21no—
38D84D411yescharacteristic
39C4×D21—21no—
40Dic42D411yescharacteristic
41C7×#67—21no—
42C7×D12—21no—
43C7×D12—21no—
44C7×D12D411yescharacteristic
45C4×D3×C7—21no—
46C7×Dic6D411yescharacteristic
47C2×C3×Dic7D412yes—
48C3×Dic14—21no—
49C3×Dic14—21no—
50#4248—22no—
51#4247—22no—
52D21⋊C4—22no—
53D3×Dic7—22no—
54C3×C7×C8—21no—
55C3×C7⋊C8—21no—
56C7×#41—31no—
57C2×C7×D4—31no—
58Q8×D7—61no—
59#1159—61no—
60#1159—62no—
61#1157—31no—
62#1157—61no—
63C2×Dic14—61no—
64C2×Dic14C2×D311yescharacteristic
65C7×SD16—61no—
66C7×D8—61no—
67C7×M16C2×D311yescharacteristic
68C22⋊Dic7—62no—
69#1142—61no—
70#1141—61no—
71C4×Dic7—31no—
72#1136C2×D311yescharacteristic
73#1007—71no—
74#947—71no—
75#908—71no—
76#908D711yescharacteristic, complemented
77C3×#101—71no—
78#825—71no—
79#825—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D3×C7—21no—
83C2×D3×C7—42no—
84C2×D3×C7—41no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—21no—
88Dic21—41no—
89C3×Dic7—22no—
90C7×Dic3—21no—
91#454—211noSylow
92C23×C7—61no—
93C7×Q8—31no—
94C7×D4—31no—
95C7×D4—61no—
96C7×D4—61no—
97C7×D4—61no—
98C2×C4×C7—61no—
99C2×C4×C7C22×D311yescharacteristic
100C22⋊D7—62no—
101C22⋊D7—61no—
102C2×Dic7—62no—
103C2×Dic7—61no—
104C2×Dic7D1212yes—
105D28—31no—
106C4×D7—62no—
107C4×D7—61no—
108Dic14—21no—
109Dic14—21no—
110Dic14—31no—
111Dic14—62no—
112C7×C8—21no—
113C7⋊C8—21no—
114C2×C3×Q8—71no—
115#239—141no—
116#239—141no—
117D3×Q8—141no—
118D3×Q8—141no—
119#235—71no—
120#235—142no—
121#235C2×D711yescharacteristic
122C2×D12C2×D711yescharacteristic
123C3×M16—71no—
124C3×M16C2×D711yescharacteristic
125(C2×D3)⋊C4—142no—
126C4×Dic3—71no—
127D24—21no—
128D24—141no—
129#204—21no—
130#204—141no—
131C2×C3×C7—21no—
132C2×C3×C7#12011yescharacteristic
133D21—41no—
134D3×C7—41no—
135D3×C7—81no—
1362−1+4—211no—
137AGL(1,ℤ/8)—31no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—211no—
141#104—211no—
142#101—71no—
143C22×C7—61no—
144C22×C7—122no—
145C22×C7—121no—
146C22×C7C2×D1211yescharacteristic
147C2×D7—61no—
148C4×C7—61no—
149C4×C7D3×D411yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—22no—
152Dic7—61no—
153Dic7—121no—
154C22×D3—21no—
155C3×Q8—141no—
156C3×Q8—141no—
157C2×C3×C4—72no—
158C2×C3×C4C22×D711yescharacteristic
159#67—21no—
160#67—141no—
161C2×Dic3—141no—
162D12—21no—
163D12—21no—
164D12—71no—
165D12—142no—
166D12C22⋊D711yescharacteristic
167C4×D3—21no—
168C4×D3—142no—
169C4×D3—142no—
170C4×D3—141no—
171Dic6—71no—
172Dic6—142no—
173Dic6C22⋊D711yescharacteristic
174C3×C8—21no—
175C3×C8—141no—
176C3×C7#45411yescharacteristic, complemented, Hall
177#41—31no—
178#41—211no—
179#41—421no—
180#41—422no—
181#41—421no—
182C2×Q8—71no—
183C2×Q8—421no—
184C2×Q8—421no—
185C2×D4—31no—
186SD16—61no—
187SD16—421no—
188D8—61no—
189D8—421no—
190M16—71no—
191M16D3×D711yescharacteristic
192C22⋊C4—422no—
193C42—211no—
194C2×C7—21no—
195C2×C7—121no—
196C2×C7—241no—
197C2×C7#88211yescharacteristic
198D7—121no—
199C22×C3D4×D711yescharacteristic
200C2×D3—21no—
201C2×D3—42no—
202C2×D3—41no—
203C2×D3—141no—
204C3×C4—142no—
205C3×C4D4×D711yescharacteristic
206C3×C4C2×C22⋊D711yescharacteristic
207Dic3—21no—
208Dic3—141no—
209Dic3—281no—
210C23—61no—
211Q8—31no—
212Q8—141no—
213Q8—141no—
214Q8—211no—
215Q8—422no—
216D4—31no—
217D4—61no—
218D4—61no—
219D4—61no—
220D4—211no—
221D4—422no—
222D4—421no—
223C2×C4—61no—
224C2×C4—72no—
225C2×C4—422no—
226C2×C4—422no—
227C2×C4—421no—
228C2×C4—421no—
229C2×C4C2×D3×D711yescharacteristic
230C8—21no—
231C8—141no—
232C7#475311yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3#650011yescharacteristic
235D3—41no—
236D3—81no—
237D3—281no—
238C22—61no—
239C22—122no—
240C22—121no—
241C22—421no—
242C22D7×D1211yescharacteristic
243C4—61no—
244C4—142no—
245C4—421no—
246C4—841no—
247C4#6668111yescharacteristic
248C4D3×C22⋊D711yescharacteristic
249C3#8997311yescharacteristic, complemented, Sylow
250C2—21no—
251C2—121no—
252C2—241no—
253C2—841no—
254C2#1062103711yescharacteristic
subgroupG/H
centerC2#10621037
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
hypercenterM16D3×D7
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10621551C2semidirect
2. C7×#908C2semidirect
3. #10620932C2semidirect
4. #10620866C2semidirect
5. #10620664C2semidirect
6. C3×#6398C2semidirect
7. #10620513C2semidirect
32. #6398D3semidirect
76. #908D7semidirect
176. C3×C7#454semidirect
232. C7#4753semidirect
249. C3#89973semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#106215512
C7×#9082
#106209322
#106208662
#106206642
C3×#63982
#106205132
#899733
#47537